All of the 3rd grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$43.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$17.00$ each for teachers and $$8.50$ each for students, and the group paid $$102.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 43.5}$ ${17x+8.5y = 102}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-17x-6y = -87}$ ${17x+8.5y = 102}$ Add the top and bottom equations together. $ 2.5y = 15 $ $ y = \dfrac{15}{2.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+3y = 43.5}$ to find $x$ ${8.5x + 3}{(6)}{= 43.5}$ $8.5x+18 = 43.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {17x+8.5y = 102}$ and get the same answer for $x$ ${17x + 8.5}{(6)}{= 102}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.